Finite Group Databasebeta

← #26103305

Subgroups — #26103305

Order 12032.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243154C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3C47×#1820C211yesmaximal, characteristic
4#15689180—21no—
5C2×C22⋊(C47⋊C8)—21no—
6C2×#13093935C411yescharacteristic, complemented
7C2×C4×C47⋊C8C411yescharacteristic, complemented
8#15688878—21no—
9#15688849C412yescomplemented
10#15688847—21no—
11C2×#117×C47C2211yescharacteristic
12C47×#381—21no—
13C47×#342—21no—
14C47×#342C412yes—
15C22×C47⋊C8—21no—
16C22×C47⋊C8—21no—
17C4⋊(C47⋊C8)—41no—
18C22⋊(C47⋊C8)—24no—
19C22⋊(C47⋊C8)—22no—
20#13093935—41no—
21#13093935—41no—
22C4×C47⋊C8—41no—
23C23×C4×C47D411yescharacteristic
24#117×C47—41no—
25#117×C47—41no—
26C2×C4⋊C4×C47C2×C411yescharacteristic
27C2×C4⋊C4×C47Q811yescharacteristic
28C2×C22⋊C4×C47—22no—
29C2×C22⋊C4×C47C2×C411yescharacteristic
30C2×C22⋊C4×C47D411yescharacteristic
31C2×C42×C47C2×C411yescharacteristic
32C2×C42×C47D411yescharacteristic
33#95×C47—41no—
34#95×C47—41no—
35C24×C47C22⋊C411yescharacteristic
36C22×C4×C47—22no—
37C22×C4×C47—21no—
38C22×C4×C47—21no—
39C22×C4×C47—41no—
40C22×C4×C47—42no—
41C22×C4×C47C4211yescharacteristic
42C22×C4×C47C22⋊C411yescharacteristic
43C22×C4×C47C22⋊C411yescharacteristic
44C22×C4×C47C4⋊C411yescharacteristic
45C22×C4×C47C4⋊C411yescharacteristic
46C22×C4×C47C4⋊C411yescharacteristic
47C4⋊C4×C47—41no—
48C4⋊C4×C47—41no—
49C22⋊C4×C47—41no—
50C22⋊C4×C47—41no—
51C22⋊C4×C47—42no—
52C22⋊C4×C47—42no—
53C42×C47—21no—
54C42×C47—21no—
55C42×C47—21no—
56C42×C47—21no—
57C2×C47⋊C8—22no—
58C2×C47⋊C8—22no—
59C2×C47⋊C8—41no—
60C2×C47⋊C8—41no—
61C2×C47⋊C8—41no—
62C2×C47⋊C8—41no—
63C2×C47⋊C8—42no—
64C23×C47—21no—
65C23×C47—21no—
66C23×C47—41no—
67C23×C47—41no—
68C23×C47#9511yescharacteristic
69C23×C47#10412yes—
70C2×C4×C47—21no—
71C2×C4×C47—21no—
72C2×C4×C47—21no—
73C2×C4×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—42no—
80C2×C4×C47—42no—
81C2×C4×C47—41no—
82C2×C4×C47—41no—
83C2×C4×C47—41no—
84C2×C4×C47—41no—
85C2×C4×C47—82no—
86C2×C4×C47—82no—
87C2×C4×C47—82no—
88C2×C4×C47#9911yescharacteristic
89C2×C4×C47#9911yescharacteristic
90C47⋊C8—42no—
91C47⋊C8—42no—
92C47⋊C8—82no—
93#7214—471nomaximal, Sylow
94C22×C47—21no—
95C22×C47—21no—
96C22×C47—22no—
97C22×C47—42no—
98C22×C47—41no—
99C22×C47—41no—
100C22×C47—81no—
101C22×C47H(3,ℤ/4)11yescharacteristic
102C22×C47#33911yescharacteristic
103C22×C47#34211yescharacteristic
104C4×C47—21no—
105C4×C47—21no—
106C4×C47—41no—
107C4×C47—41no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—41no—
111C4×C47—162no—
112#1820D4711yescharacteristic
113#1750—471no—
114#1464—471no—
115C2×C47—21no—
116C2×C47—21no—
117C2×C47—42no—
118C2×C47#128411yescharacteristic
119C2×C47#139111yescharacteristic
120C2×C47#139511yescharacteristic
121C2×#117C2×D4711yescharacteristic
122#432—941no—
123C2×C22⋊C8—941no—
124C2×#97—471no—
125C2×C4×C8—471no—
126#381—21no—
127#342—21no—
128#342Dic4712yes—
129#336—941no—
130#324—472no—
131#323—941no—
132C47#721411yescharacteristic, complemented, Sylow
133C23×C4C22⋊D4711yescharacteristic
134C22×C8—941no—
135C22×C8—941no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4Dic9411yescharacteristic
139C2×C4⋊C4C4×D4711yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4C2×Dic4711yescharacteristic
142C2×C22⋊C4C22⋊D4711yescharacteristic
143C2×C42C4×D4711yescharacteristic
144C2×C42D18811yescharacteristic
145C4⋊C8—1881no—
146C22⋊C8—944no—
147C22⋊C8—942no—
148#97—1881no—
149#97—1881no—
150C4×C8—1881no—
151#95—41no—
152#95—41no—
153C24C22⋊Dic4711yescharacteristic
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4C4×Dic4711yescharacteristic
160C22×C4Dic47⋊C411yescharacteristic
161C22×C4Dic47⋊C411yescharacteristic
162C22×C4C4⋊Dic4711yescharacteristic
163C22×C4(C2×D47)⋊C411yescharacteristic
164C22×C4(C2×D47)⋊C411yescharacteristic
165C2×C8—942no—
166C2×C8—942no—
167C2×C8—1882no—
168C2×C8—1881no—
169C2×C8—1881no—
170C2×C8—1881no—
171C2×C8—1881no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23C47⋊#9511yescharacteristic
187C23#1309395112yes—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4#1309393811yescharacteristic
207C2×C4#1309393811yescharacteristic
208C8—3762no—
209C8—1882no—
210C8—1882no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22#1568887911yescharacteristic
219C22#1568888211yescharacteristic
220C22#1568888911yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—162no—
229C2—21no—
230C2—21no—
231C2—42no—
232C2#2024124011yescharacteristic
233C2#2024150811yescharacteristic
234C2#2024151611yescharacteristic
subgroupG/H
centerC22#15688889
commutatorC22×C4×C47C42
FrattiniC2×#117C2×D47
FittingC47×#1820C2
socleC22×C47#342
prime radicalC24×C47C22⋊C4
cosocleC2×#117×C47C22
hypercenter#1820D47
hypocenterC47#7214
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×#13093935C4semidirect
7. C2×C4×C47⋊C8C4semidirect
9. #15688849C4semidirect
132. C47#7214semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202431542
#202417222
C47×#18202
#721447